If you follow THH's counsel, you will not need to explain non-trivial results such as this one. I would not be surprised if your problem description would become much more manageable and compact if you were to follow his advice.
Eric I still think I motivated why I would like to include the definition as is, but i also see a value to add what I think THHuxleynew's is indicating. I'm still not convinced that his suggestion can precisely lead to the generalisation I'm looking for to include as well in as suplementary question.